Reflective metalens with sub-diffraction-limited and multifunctional focusing

We propose an ultra-thin planar reflective metalens with sub-diffraction-limited and multifunctional focusing. Based on the equal optical path principle, the specific phase distributions for multifunction focusing are derived. Following the formulas, on-center focusing with the characteristics of sub-diffraction-limited, high focusing efficiency (85%) and broadband focusing is investigated in detail. To demonstrate the flexibility of the reflective metalens, off-center and dual spots focusing (at the horizontal and longitudinal directions) are demonstrated. Note that all these focusings are sub-diffraction-limited due to the evanescent-filed enhancement mechanism in our elaborately designed structure. The designed reflective metalens will find important applications in super-resolution imaging, microscopes, and spectroscopic designs.

Scientific RepoRts | 7: 12632 | DOI: 10.1038/s41598-017-13004-z off-center, and dual-spots focusing (at horizontal and longitudinal directions) are exhibited. For the on-center focusing, the sub-diffraction-limited and broadband focusing characteristics are investigated in detail. Finally, the dependence of N.A. on the focusing area is discussed.

Results
Designs and structure. Figure 1a shows the schematic of the reflective metalens. It is composed of amorphous silicon (n = 3.6) nanoblocks array and a gold ground plane with a dielectric spacer (n = 1.46) between the two layers. The side view of the unit cell is shown in Fig. 1b, with the heights from top to bottom are h = 400 nm h 1 = 200 nm h 2 = 150 nm, respectively. Top view of the unit cell is shown in Fig. 1c, from which we can see that the nanoblock's width W = 132 nm, length L = 600 nm and the lattice constant P = 650 nm. The required phase can be imparted by rotating the nanoblock with an proper angle θ. In our proposed structure, the parameters of the To achieve a reflective metalens with high focusing efficiency, the nanoblocks should perform as a half-waveplate which is able to convert the circular polarized light into reflected light with opposite helicity. Here the nanoblock is selected for its simple structure and strong polarization conversion effect. Similar to other nanostructures with azimuthal asymmetry, nanoblock can also exhibits form birefringence 33 . A single nanoblock resembles a channel waveguide, which will have a corresponding effective refractive index (n eff ) for the two orthogonal linear polarization states, E x and E y . For the channel waveguide that owns a circular cross-section, the neff for both linear polarized light (E x and E y ) are the same. However, for our designed nanoblock that owns a rectangle cross-section, the effective refractive index neff for E x and E y is different, which is equivalent to form birefringence 34 . Therefore, under circularly polarized incident light, the nanoblock function as a waveplate and high polarization conversion efficiency can be achieved. Figure 2a,b shown the reflectance and conversion efficiency of the nanoblock as a function of the incidence wavelength, respectively. It can be observed that both the reflectance and conversion efficiency reach their peaks at the designed wavelength (λ = 1550 nm). In this case, the reflectance reaches 96% with a polarization conversion efficiency as high as 99.6%. Here, the polarization conversion efficiency is calculated as the ratio of the reflected power with opposite helicity to the total reflected power. According to the definition in ref. 27 . which is defined as the ratio of the reflected power with opposite helicity to the total incident power, the polarization conversion efficiency should be 95.6%. The phase shift of the nanoblock with various rotation angle θ are plotted in Fig. 2c. It can be observed that the phase shift and the rotation angle satisfy the condition of Pancharatnam-Berry (P-B) phase (ϕ = 2θ), from which the phase shift coverage of 0 to 2π is obtained 27,35 . The dependence of phase shift on the rotation angle θ, for various incident wavelengths, is shown in Fig. 2d. For a wavelength range of 200 nm (from 1.46 μm to 1.66 μm), the phase shift can achieve a coverage of 2π, indicating the broadband characteristic of our designed nonablock.
In order to focus an incident plane wave, a phase compensation mechanism is required at the surface of metalens. The corresponding phase profile ϕ(x, y) of the metalens follows the equal optical path principle 6,16 : where λ is the incidence wavelength, x and y are the coordinates of the nanoblocks, and f is the designed focal length. The required phase is imparted based on the P-B phase via rotating the nanoblock with an angle θ(x, y) = ϕ(x, y)/2. Hence, each nanoblocks at coordinate (x, y) should be rotated with an angle

Discussion
Reflective metalens with on-center focusing. As illustrated in Fig. 3, the reflective metalens is designed to realize on-center focusing for RCP normal incident light at the wavelength of 1550 nm. The focal length is designed to be f = 20 μm, whereas the concept is scalable to any values. From Eq. 2, the rotation angle of the nanoblocks on the x axis is plotted in Fig. 3a. Top view of the reflective metalens is shown in Fig. 3b, where the targeting phase in Fig. 3a is imparted by rotating the nanoblock with a proper angle. Figure 3c,d show the simulated focal spot intensity (|E| 2 ) profile at x-y and x-z planes, respectively. The plane metalens provides a strong focusing capability with a N.A. of ~0.65 and a focusing efficiency up to 85%. The focusing efficiency is defined as the fraction of the incident light that pass through a radius equal to three times of the FWHM spot size 25 .
Sub-diffraction-limited characteristic of the focusing spot. The corresponding vertical cut of the focal spot is depicted in Fig. 3e. The full width at half maximum (FWHM) of the focal spot is 1136 nm (less than λ/2 N.A.), which indicates a sub-diffraction-limited focusing of our reflective metalens 27 . This effect can be attributed to our elaborately designed metalens structure. The gold ground plane is introduced to enhance the evanescent waves by the excitation of the surface plasmon. The nanoblocks arrays function as a coupler, which will convert the enhanced evanescent components into propagating waves 30 . With these two conditions, a metalens with sub-diffraction-limited focusing in the far-filed can be achieved. To verify the evanescent-filed enhancement mechanism of our designed metalens, the gold ground plane is replaced by a perfect electrical conductivity (PEC) ground plane. Without the enhanced evanescent waves by the excitation of the surface plasmon, the corresponding FWHM increased to 1210 nm (larger than λ/2 N.A.), suggesting the diffraction-limited characteristic of the focusing spot. The metal-dielectric-metal (MIM) configuration generally also have high-efficiency for wavefront control 36 . To exhibit the advantage of our designed dielectric-dielectric-metal (DDM) configuration, a MIM configuration for focusing at the designed wavelength (1550 nm) is also taken into consideration. The Si nanoblock is replaced by Au nanoblock with the other configuration the same as Fig. 1  the reflectance and polarization conversion efficiency of the nanoblock as a function of the incidence wavelength, respectively. It can be observed that the reflectance is 89.5% and the polarization conversion efficiency is 96.6% at the designed wavelength (λ = 1550 nm). According to the definition in ref. 27 . which is defined as the ratio of the reflected power with opposite helicity to the total incident power, the polarization conversion efficiency should be 86.5%. The reflective metalens constructed of the Au nanoblocks is also demonstrated to realize on-center focusing. Figure 4c shows the simulated focal spot intensity (|E| 2 ) profile at x-z plane. It is obvious that a focusing spot is exhibited at the designed focal length (f = 20 μm). The N.A. is ~0.69 and the simulated focusing efficiency is 73% at the designed wavelength. The corresponding vertical cut of the focal spot is shown in Fig. 4d, in which the FWHM of the focal spot is 1102 nm (less than λ/2 N.A.), indicating that the reflective metalens constructed of the Au nanoblocks also overcomes the diffraction limit. Therefore, despite reflective metalens with MIM configuration can also achieve diffraction-limited focusing, the focusing efficiency is less than that with DDM configuration.
As mentioned above, by introducing the P-B phase, the designed nanoblock owns broadband characteristic (can achieve a phase coverage of 2π among a broad wavelength range). Hence, the metalens that constructed by the nanoblocks is bound to exhibit broadband focusing effect. Figure 5a-c show the simulated focal spot intensity profile at x-z plane for wavelengths λ = 1.46 μm, 1.56 μm, 1.66 μm, respectively. The focal length as a function of the incident wavelength is shown in Fig. 5d. From Fig. 5a-d, it can be seen that the designed reflective metalens exhibits focusing effect within a broad bandwidth and the focal length decreases as the increment of the wavelength. These results will provide helpful guidelines in modulating the focal length.
Reflective metalens with off-center focusing. To focus the incident light to an arbitrary position A(x 1 , y 1 , f), each nanoblocks at coordinate (x, y) should be rotated with an angle where f 1 is the focal length, which defined as the distance from the focal point to the center of the nanoblocks plane. Based on Eq. 3, the rotation angle of the nanoblocks along the x axis are plotted in Fig. 6a. The off-center focusing spot is set at the location (3 μm, 3 μm, 20 μm). Figure 6b,c show the simulated focal spot intensity profile at x-y and x-z planes, respectively. The focal spot shows a slight shift from the expected location (3 μm, 3 μm, 20 μm), which results from imperfect phase change imparted by rotation the nanoblock. Besides, the simulated focusing efficiency is 85% at the designed wavelength. The corresponding vertical cut of the focal spot is shown in Fig. 6d, where the FWHM of the focal spot is 1170 nm (less than λ/2 N.A.), indicating that the off-center focusing is also overcomes the diffraction limit. Therefore, based on such a design principle, sub-diffraction-limited focusing at arbitrary position can be achieved, which will largely broaden its practical applications in laser-based microscopy, imaging and spectroscopy.
Reflective metalens with dual spots focusing. The reflective metalens with dual spots focusing at the horizontal direction is demonstrated. The required rotation angle of the nanoblocks at coordinate (x, y) is expressed as where ±x 1 represents the locations of the dual spots. In this case, the two focal spots are located at (−8.4 μm, 0 μm, 20 μm) and (8.4 μm, 0 μm, 20 μm), respectively. The rotation angle of the nanoblocks along the x axis is shown in Fig. 7a, where the curve exhibits dual parabolic shape. Figure 7b,c show the simulated focal spot intensity profile at x-y and x-z planes, respectively. It can be observed that the focal spots occur at the expected locations. The simulated results also indicate that the two focal spots own equal focusing efficiency (42%) and the N.A. decreased to 0.38. Moreover, the FWHM of the two focal spots is shown in Fig. 7d. It can be observed that the FWHM of the focal spot increases to 1980 nm (less than λ/2 N.A.), indicating the sub-diffraction-limited characteristic of the focusing spot. Hence, dual spots sub-diffraction-limited focusing can be realized at the horizontal direction and such a design principle can be further applied to realize multi-spots focusing.
To further explore the functionality of the reflective metalens, dual spots focusing at the longitudinal direction is demonstrated as well. Similarly, the required rotation angle of the nanoblocks at coordinate (x,y) should be expressed as  direction, the calculated N.A. can reach 0.84 for the lower focal spot (f = 5 μm), which is larger than most reported metalens 16,27,28 . These results will provide helpful guidelines in designing metalens with high focusing properties. In summary, an ultra-thin planar reflective metalens with sub-diffraction-limited and multifunctional focusing has been investigated. Based on the principle of equal optical path, the formulas of the requiring phase distributions for multifunction focusing are derived in detail. Following the formulas, on-center, off-center and dual spots focusing (at horizontal and longitudinal directions) are demonstrated. It worth noting that all these focusings are sub-diffraction-limited due to the evanescent-filed enhancement mechanism in our designed metalens. Moreover, the N.A. dependence on the area of the metalens is discussed. With such a design principle in our reflective metalens, one can obtain the sub-diffraction-limited focusing at any positions with high focusing properties. These results will provide helpful guidelines in designing super-resolution light imaging and sensing systems.

Methods
Simulations. The performance of the the proposed metalenses are characterized by using the three-dimensional finite difference time domain (FDTD) method from Lumerical Inc. For the simulation of the unit cell, periodic boundary conditions are applied along the x and y axis and perfectly matched layers(PML) is applied along the z axis. For the simulation of the metalenses, PML are applied along the three axis for the specific phase elements of the designed metalens. The simulated total area of the metalens is 33.8 × 33.8 μm 2 with 53 × 53 unit cells.